Crossing the Bridge Training #2

Vertical Alignment: Secondary Mathematics

Golden ApplesA prince picked a basketful of golden apples in the enchanted orchard. On his way home, he was stopped by a troll who guarded the orchard. The troll demanded payment of one-half of the apples plus two more. The prince gave him the apples and set off again. A little further on, he was stopped by a second troll guard. This troll demanded payment of one-half of the apples the prince now had plus two more. The prince paid him and set off again. Just before leaving the enchanted orchard, a third troll stopped him and demanded one-half of his remaining apples plus two more. The prince paid him and sadly went home. He had only two golden apples left. How many apples had he picked?

Essential Questions• Why should a teacher be concerned with the

vertical alignment of standards in his/her content area?

• How are the Common Core State Standards for Mathematics (CCSSM) organized?

Learning Targets• I can name the two types of standards outlined in the

Common Core State Standards for Mathematics (CCSSM).

• I can evaluate the progression of standards within a domain.

• I can list the 8 Standards for Mathematical practice.• I can explain the organization of the Standards for

Mathematical Content.

Common Core State Standards for Mathematics (CCSSM)

WHAT ARE THE STANDARDS FOR MATHEMATICAL

PRACTICE?

The Origin of the Mathematical Practices• National Council of Teachers of Mathematics

process standards: problem solving, reasoning and proof, communication, representation, and connections.

• National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition.

NCTM Process Standards• Problem-Solving• Connections• Reasoning and Proof• Communication• Representation

NCTM Process Standards• Problem Solving: using mathematical concepts and

processes to answer questions• Reasoning and Proof: justifying mathematical ideas• Communication: explaining mathematical thinking in

clear mathematical language• Connections: relating mathematical topics to

each other as well as to other disciplines

• Representations: illustrating mathematical ideas in multiple formats

CCSSM Mathematical PracticesThe Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to NCTM’s Mathematical Processes from the Principles and Standards for School Mathematics.

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels

should seek to develop in their students. These practices rest on

important “processes and proficiencies” in mathematics

education.” (CCSS, 2010)

CCSS Standards for Mathematical Practice1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of

others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

NCTM Process Standards CCSS Mathematical Practices

Problem Solving

Reasoning and Proof

Communication

Connections

Representations

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and

critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in

repeated reasoning.

1. Highlight the verbs that illustrate the student actions for Practice #3: Construct viable arguments and critique the reasoning of others.

2. Discuss the following questions with a partner: What would you see happening in the classroom based on your

examination of this standard? What impact does this standard have on your current teaching practices? What evidence/artifacts will show that students are developing this habit

of mind using this practice?

3. Illustrate this practice in your small group.

A Closer Look at Mathematical Practice #3

Common Core State Standards for Mathematics

WHAT ARE THE STANDARDS FOR MATHEMATICAL CONTENT AND

HOW ARE THEY ORGANIZED?

Design and Organization of the Common Core State Standard for Mathematics

Standards for Mathematical Practice• Carry across all grade levels• Describe habits of mind of a mathematically expert studentStandards for Mathematical Content• K-8 standards presented by grade level• Organized into domains that progress over several grades• Grade introductions give 2–4 focal points at each grade

level• High school standards presented by conceptual category

(Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)

Common Core FormatHigh School

Conceptual Category

Domain

Cluster

Standards

K-8

Grade

Domain

Cluster

Standards

(No pre-K Common Core Standards)

Common Core FormatDomains are larger groups of related standards.

Standards from different domains may sometimes be closely related.

Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

Standards define what students should be able to understand and be able to do.

Grade Level/Course Overview

K 1 2 3 4 5 6 7 8 Counting

& Cardinality

Ratios & Proportional Relationships

Operations and Algebraic Thinking The Number System

Number and Operations in Base Ten Expressions and Equations

Fractions Functions

Measurement and Data Statistics and Probability

Geometry Geometry

Domains K-8

Vertical Alignment: Charting Content Progressions

• Carefully consider the example that follows.

• With your table companions, create a chart which depicts a progression of the content standards for one of the other conceptual categories.

• At this point, let’s use the time we have to make our charts as specific as possible with the information we have.

Focus on UnderstandingThe Standards for Mathematical Content are a balanced combination of procedure and understanding.

Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content.

Extended Content Standards

The No Child Left Behind Act requires that all students, including those with the most significant cognitive disabilities, have access to the standard course of study at grade level. The Extended content standards provide entry point extensions so that all students have meaningful and functional access to grade level standards. These Standards should be used to develop goals, learning experiences and materials for students with the most significant cognitive disabilities.

Common Core and Extended Common Core Crosswalks and Unpacked Standards

http://www.ncpublicschools.org/acre/standards/common-core-tools/

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

http://www.ncpublicschools.org/acre/standards/extended-tools/

Closing Activity• Based on the session today list three (3) things

that you learned that are different or that will be new for you next year in Mathematics.

• List two (2) things that you can begin working on right now.

• List one (1) thing that you would like to know more about.

Contact InformationDawn Perks

Grayling B. Williams,Secondary Mathematics Specialists

dawn.perks@dpsnc.net grayling.williams@dpsnc.net

(919) 560-2000(Ext. 21314)